1

11 Chapter 4 .

I11 THE AERODYNAMIC DESIGN OF AXIAl-FlOWi¡

1I AND RADIAl-INFlOW TURBINES! ¡ I!

iI

By DR. A. DOUGLAS CARMICHAEL

ADV ANCES in the performance of axial-flow and (3= Relative angle measured from axial direction, de-I radial-inflow turbines have come from the application of greesi basic fluid-dynamic principies. At present, the flow condi- '"{= Isentropic exponent.¡ tions in these machines are not completely understood be- <5 = Deviation angle, degrees

cause of the complexity of their geometries and because of ~= Differencereal-fluid effects. However, considerable progress has been e = Diffuser recovery factormade with use of experimental data to provide design infor- 11= Isentropic efficiencymation where the fluid flow conditions cannot be adequate- e = Tangential direction, degreesIy analysed. A = Work-speed parameter, equation 45

In this chapter an attempt has been made 10 collect and v= Kinematic viscosity, ftsummarize important theoretical and experimental data ~= Loss coefficientthat can be used in design of axial-flow and radial-flow tur- p = Derisity, Ibm/cu ftbines. Comparison between the performance of these two 1/1= Tangential force coefficient, equation 71types of turbines is not discussed.

SubscriptsNOMENCLATURE i= Initial r

is= IsentropicAxial Turbines N= Nozzle

a= Velocity of sound, ft/sec 0= Stagnationb= Blade axial chord, ft p= Polytropicc= Blade chord, ft R = Relative

cp = Specific heat at constant pressure, Btu/lbm R R = Rotor

cv= Specific heat at constant volume, Btu/lbm R ts= Total (or Stagnation) to staticD= Hydraulic mea n diameter, equation 54, ft e = Tangentiale= Specific interna! energy, Btu/lbm 1 = Ahead of turbomachine

go = Constant in Newtons Law, ft Ibm/lbf sec2 1 = Ahead of rotorh= Specific enthalpy, Btu/lbm 2;= Downstream of turbomachineh = Blade height, ft 2 = Downstream of rotori= Incidence angle, degrees SuperscriptsJ = Energy conversion factor (778.16), ft Ibf/Btu * - BI d d.t .k - CI ft - a e con I lons- earance T k. f bl d h. kM ' / . = a Ing account o a e t IC nessm = ass flow, Ibm sec

M = Mach numbern= Distance along a normal, ft Radial-lnflow Turbines

N= Rotational speed, rpm a = Velocity of sound, ft/sec0= Opening or throat width, ft a, b = Constants in equation 83P= Pressure, Ibf/sq ft A = Area, sq ftQ= Heat transferred, Btu/lbm b = Tip width of the rotor, ftr= Radius, ft EFG = Constants in equation 85r c = Radius of curvature, ft ' ~o = Constant in Newtons Law, ft Ibm/lbf sec2

R = Specific gas constant, ft Ibf/lbm R J= Energy conversion factor (778.16), ft Ibf/BtuRe= Reynolds number m = Mass flow, Ibm/sectR = Reaction m = Constant in equation 92s= Blade pitch, ft n = Distance along a normal, ft

¡' s= Specific entropy, Btu/lbm R N = Rotational speed, rpmT= Temperature, R Ns = Specific speedu = Peri pheral velocity, ft/sec o = Opening or throat width, ft IV= Absolute velocity, ft/sec P = Pressure, Ibf/sq ft~= Relative velocity,.ft/sec. q = Volume flow, cu ft/sec I

~ W= Shaft work, Btu/lbm r = Radius, ftx= Axial distance, ft rx = Radius at end of slip region, ftx= Distance along surface, ft T= Temperature, R

: a= Absolute angle mea su red from axial direction, u = Peripheral speed, ft/sec¡ degrees V= Absolute velocity, ft/sec

62

~

W = Relative velocity, ft/sec W is the rate of output shaft work per unit mass

z = Number of blades flow

a = Absolute angle measured from axial direction, e is the specific interna! energy of the fluid

degrees P/pJ is the flow work associated with moving the fluid(3 = Relative angle l"Qeasured from axial direction, V2/2goJ is the kinetic energy of the fluid

degre~s o' . A u~eful property of the fluid termed enthalpy, h can be'Y = Isentroplc exponent d f '

dA h h lo o elne-E = ngle between t e tangent to t e stream Ines In

the ~eridional plane and the impeller axis of h == e + ~ (2)

rotatlon pJ~ = Slip factor for rotar . o .77 = Isentropic efficiency Th~ sum of klnetlc energy aijd enthalpy IS termed the stag-v = Kinematic viscosity, ft2/sec natlon enthalpy, ho, where-

w = Angular speed, radians/sec ~

ho == h + - (3)Subscripts 2goJ

is = Isentropic

m = Meridional The steady flow energy equation can now be rewritten as-

o = Stagnation o .

opt = Optimum Q - W = ho2 - ho! (4)

S = Stator In a turbomachine the heat transfer is small ti.e. the process

R = Rotor is nearly adiabatic) thus it is customary to neglect the heat

e = Tangential transferred to the fluid, Q and therefore-

0,1,2,3,4, = Stations through the turbine

W = hO1 - hO2 (5)

Superscripts

* = Blade angle This equation indicates that shaft work delivered by the

turbomachine isequalto the decrease in stagnation enthalpy

Sign Convention: Absolute and Relative angles and veloci- across the machine in the absence of external friction

ties are positive in the direction of rota- torqueso .tion. En trop y. A further useful property of the fluid termed

entropy, s, is derived from the second law of thermodynam-BASIC THERMODYNAMICS AND FLUID MECHANICS ics. For a reversible process-

-A ~ummary ~f basic thermody~amic and fluid ~e~hanic - ( dQ) -,- equatlons used In the study of axlal-flow and radlal-lnflow ds == - (6)

turbines is presentedo No proofs or deriva!ions are given T

since these can be found in good texts on the subjectso This

section is similar, although not identical, to the basic therm- Entropy is related to the other thermodynamic properties

odynamics and fluid mechanics given in the previous through the expression-

chapter.U . P ( 1) 1 nlts Tds = de + - d - = dh - - dP (7)

Engineering units are used throughout this chapter; all J p pJ

forces are expressed in Ibf units, the unit of mass is the Ibm

and temperatures are in degrees R, The constant go in New- Th¡s expression is valid for any process in the absence of

ton 's second law is used in all equations concerned with capillarity, chemical reaction, electricity and magnetismo

kinetic energy and change of~entum; go has the units PERFECT AND SEMI-PERFECT GASES, The perfect

ft Ibm per Ibf sec2 o and semi-perfect gas have the equation of state-

Thermodynamics PSTEADY FLOW ENERGY EQUATION. The steady - = RT (8)

flow energy equation can be derived from the first law of p

thermodynamics and can be applied to any continuous flow R .o o h . Th f where IS the gas constant.

process such as expanslon In a turbomac Ineo e sum o o o

work done and heat supplied to the fluid in a turbomachine Ispeclflc heat ds f~t Cdonstant pressure, cp, and constanto vo ume C are e Ine as-between statlons 1 and 2 can be express as- , v'

Q-W= (e2 +~+ ~ ) -(e! +~+~ ) C ==/~\P2J 2goJ \ P.J 2goJ p \aT/p

(1) (9)where (a )Q is the rate of heat transfer to the fluid per unit c v == ~

mass flow aT v

63

.For a perfect gas both cp and Cv are constant while for a POL YTROPIC EFFICIENCY. Another concept of effi-semi-perfect gas both cp and Cv are functions of tempera- ciency is the small stage or polytropic efficiency. Polytropictu re and can be found from gas tables. For a perfect gas the efficiency is the limiting value of isentropic efficiency asfollowing relationship's may be definedor derived: the pressure change in the expansion process approaches

zero. Polytropic' efficiency can be determined for a perfectc gas from equation 16 as-

'" - p1 = - (10) T

oCv -d-

..," = To (17)"p

R 'Y-1c - c = - (11) ~ dPo)- P v J 1- 1+- 'Y

Po

( 'Y ) R thus-c = - - (12)p 'Y-1 J 'Y-1

. ~ =(~) 11P1 (18)The isentropic expansion of a gas is given by the following T 01 \POI

expression:'Y Isentropic efficiency can be related to polytropic efficiency

(!-.!.-\ =( !!.2-)'Y = (!!-)-;y-:-1 (13) through the expression-

P2 J P2 T 2 'Y - 1

(P02 ) 1/P1 ISENTROPIC EFFICIENCY. As a consequence of the 1 - Psecond law of thermodynamics it can be stated that the 11 = -- 01 - 1 (19)

ideal adiabatic expansion occurs at constant entropy. This . - (!:!!3-)..r.- permits the definition of isentropic efficiency, 1/, as- 1 P 'Y

01

Actual Work COMPRESSIBLE FLOW EQUATIONS. The following1/ = . W k (14) one-dimensional equations can be determined for a perfect

- I sentroplc or -gas:

Mach Number-This may be written in terms of the change in stagnationenthalpy across the expansion process from equation 5 as- V V

M == - = (20)(hol -h02) a y "YgoRT

1/ = (15)(hol - h02,iS)

Stagnation Temperature-An Entropy - Enthalpy diagram illustrating the adiabaticexpansion is illustrated on Fig. 1. V2

To == T + (21)2goJcp

To 'Y-1, - = 1 + - M2 (22)T 2

Stagnation Pressure-this is'defined as th~ pressure of a mov,p ing fluid when it is brought to rest isentropically-o,

~

;

~ 'Yc ( - .. Po 'Y-1 2 1

Fig. 1. Entrophy- -= 1 + -- M ) 'Y- (23)Enthalpy Diagram P 2for a Turbine

EnltoP1 sContunuity Equation-

For a perfect gas-

T011--TI 'Y+111 =~(P)o~ ] (16) ~-y~ = M (~\1/2 ~ +~ M2);~)1 - ~ - 'Y PoA R ) \ 2

POI (24

64

These one-dimensional equations for compressible tIow are The continuity equation is-

very useful in the design of turbomachinery and are often

tabulatedforair. a(p,Vr) a(p,vx)+ =0 (32)

a, axFluid Mechanics

CONTINUITY. Mass conservation for uniform, steady' BOUNDARY LAYERS. It has been observed that the

flow can be expressed as- main effect of viscosity is restricted to a region clase to

V - salid surfaces; this regían is termed the boundary layer.p A cos (X - Constant (25) Outside the boundary layer the effect of viscosity can be

o I . V neglected and the inviscid equations presented above can bewhere (X iS the angle between the ve oclty and the normal used. In general the boundary layer growth along any sur-

to the area A. . o face is governed by the Reynolds number (Vx/v) and the

~ELOCITY ~RIANGL.ESo oVeloclty trlan.gle.s are used to gradient of velocity (dVldx). At low Reynolds number, also

d.efloe the veloclty an~ dlrectlon of the flUid in b~th rela- when the velocity outside the boundary layer (the free-

tlve and absolute coordlnate systems as shown on Flg. 2. t I ot ) . odl IdV/d . " t Ods ream ve OCI y Increases rapl y x IS pOSI Ive an' """,~:~ large) the boundary layer is laminar. At fligh Reynolds

~:I~~:~;. v, ~ number and also when the free-stream velocity decreases

the boundary layer becomes turbulento In regions where the

free-stream velocity is decreasing it is possible that the- vx, w, v, boundary layer may separate from the salid surface and

Rotational Velocily u, t ::=~.. ~I ~, give rise to large losses in stagnation pressureo The boundary

v layers in turbo-machines are not easy to analyze beca use of

.'

F. 2 V 1 it T o les the complicated three-dimensional nature of the flow. How-Igo o e oc y rlang

ever, two-dimensional analyses give a fair indication of true

WORK EOUA TION. The sum of the moments of .']11 the boundary layer conditions in turbomachines and are nor-

external forces is equal to the rate of change of angular mally used to provide approximate design limitations.

momentum of the system. Thus the torque delivered by a INEstream tube element of a turbine rotar to change tangential AXIAL FLOW TURB S

velocity from Vel to Ve2 between radii '1 and '2 respec- .tively is- Turblne Performance

TURBINE CHARACTERISTICS. Fromdimensional anal-Torque = rl Vo 1 -'2 Ve2 (26) ysis considerations it can be shown that similar Mach num-

Mass flow go bers and fluid flow directions are maintained in the blading of

a turbine-usirig a particular perfect or near perfect gas at

The power delivered by the element of the rotar is there- constant values of (m ~ /Po 1) and (NI..¡-f;;-; ). Similar

fore- boundary layer conditions occur when Reynolds numbers

are constant. In general the influence of Reynolds number

T J}" V is of secondary importance and performance characteristics. w x orque u 1 "e 1 - U2 e2 .w = = (27) 1],and(poI/P02)arenormallydrawnlntermsofJ x Mass flow g oJ

(m~/PoI )This equation is sometimes termed the Euler Work Equa-

tion and may be rewritten from equation 5 as-

UI VOl - U2 Ve2 1

ho 1 - h02 = (28) ~

gol g

EOUATIONS OF MOTION IN AXISYMMETRIC ~

CYLINDRICAL CO-ORDINATES. The equations for an f. -" -inviscid fluid in axisymmetric cylindrical co-ordinates in the .!

absence of body forces where , is the radial co-ordinate, e is '

the angular co-ordinate, and x is the axial co-ordinate are

given by Horlock (1) as-

avr avr VO2 goap t 5V r - + V x - - - = - - - (29) ~

ar ax r p a, '--'.

~I~

a~ a~ V~ ~2V r --.! + V x ~ + ~ = o (30) ~.I~.a r a x r 10 20 3~

Fig. 3. An Example StaQna,;on P,...u'. Aa';., "o1.!P..of Axial-Flow Tur-a V x a V x go ap bine Characteristics

Vr - + Vx - = - - - (31) for a Two-Stagea, ax p ax Turbine

65

and (N/ ~). A characteristic for a two stagQ turbine is for a streamline at constant radius. If the axial velocity V xshown on Fig. 3. In this example (my-r;;-;/PoI )and 1} are does not change across the rotor then equation 33 may beplotted against stagnation R!essure ratio with (N/~) as rewritten as-parameter. Other forms of turbine characteristics are also u V (tan a - tan a )used; see Fig. 16 for a radial irtflow turbine. ho I - h02 = Mo = x 1 2 (34)

VELOCITY OIAGRAMS. The station numbers for a sin. golgle stage turbine together with the velocity diagrams areillustrated on Fig. 4. and since tan {31 = tan al - u/V x and tan (32 = tan a2 -

u/V then ;Stalion Numbo,s 2 x

o

uVX(tan{31 - tan{32)ilho = -- (35)

gol

ABSOLUTE ANO RELATIVE ENTHALPIES. Theenthalpy conditions in an axial-flow turbine stage can bewritten as follows:

1. Upstream of the rotor blade rowa. absolute

NOZZLES vlC\ (\ hol =hl +- (36)~ ", 2gl)J

ROTaR h --j5J ~ °J- -:;¡-;., Fig. 4. Geometry b. relative to the rotor../-'" /-"" - w. u V2 v:x:, and VeloCity Tri-

1- - I ~ I_) angles for an Axial- W2 W2 - ~

--J ... Flow Turbine I 1 1

w.,(-) hO1 R = h1 + -=hO1 + (37), 2goJ 2goJ

Several assumptions, associated with the blockage effectof the blade thickness, have been used in drawing thevelocity triangles. Three such methods in current use are as 2. Oownstream of the rotor blade rowf 11 a. absoluteo ows: -

1. The stations selected are between the blade rows sothattheeffectoftrailingedgethicknessisneglected h =h + ~= - U(Vel - Ve2) (38)

except perhaps as a loss in stagnation pressure. 02 2 2g J ho 1 g J2. The stations selected are at the trailing edges of the o o

blade rows and the blockage effect of the trailing .edges is included in the continuity equations. b. relatlve to the rotor

3. Separate velocity triangles are drawn for the exit of W~ u~ - u¡one blade row and the entrance of the following h02,R = h2 + - = ho l,R + = ho l,Rblade row. The blockage effect of the trailing edges 2goJ 2goJand leading edges is included. (39)

There are discrepancies of the fluid flow angle when OEGREE OF REACTION. Reaction, ~,is a useful termthese three methods are compared which can be five to ten in turbines which describes the form of velocity changes indegrees for blades having thick leading and trailing edges. the blade rows. It is defined as the ratio of the change inAlthough it is not possible to say which of the methods static enthalpy to the change in stagnation enthalpy acrossoutlined above is superior, experience has shown that some the rotor or-difficulties are encountered with the first method (wherethe velocity triangles are drawn between the blade rows) be- hl - h2cause it is possible to draw velocity triangles which have «f{ = (40)subsonic velocity components while in the blading the hol - h02

blockage of the trailing edges results in sonic or supersonicvelocity components. The exhaust duct conditions down. For a rotor having constant radius equation 40 can be writ-stream of the last turbine blade row would be based on ten in terms of the rotor relative velocities and stagnationflow without trailing edge blockage for all methods of enthalpy rise as-drawing the velocity triangles.

WORK EQUATION. The drop in stagnation enthalpy W2 - W2across an axial turbine (along a streamline) is obtained from «f{ = 2 I (41)

equation 28 as- 2goJ ilho

h - h = U I Ve I - U2 V 02 Equation 41 shows that for a zero reaction (or impulse con.01 02 g J dition) the exit relative velocity, W2, is equal to the inlet

o relative velocity, W l. Experience has showri that turbine

V -:v ) performance deteriorates when the reaction is negative.= u ( el 82 (33) For a turbine stage having V x unchanged across the

gol rotor equation 40 can be manipulated to give-

66

-",

WOI + WO2If{ = - (42)

2u

and ha

$'V O + VO.,If{ = 1 - l - (43) R

2 u . R

For a stage having zero tangential velocity at exit-

P,

VOl hIf{ = 1 - - (44) I

2 u .c h.,

WORK-SPEED PARAMETER. In addition to reaction > ho 2

there is another type of parameter which can be used to ~cdefine the stage loading. Stewart (2) defined a work-speed x1- h

parameter- ~ h2

U2>.. == (45)

gol ~ ho

Similar parameters serving the purpose of defining the stageloading used by other workers are- ENTROPY .

Fig. 5. Entropy-Enthalpy Diagram for an Axial-Flow Turbine.gol ~ho 2goJ ~ho u and u-~' U2 . /- fA L

g J~h -/2- fA Lg J~h. andtherotorlosscoefficientis-V 50J "" FIO V L50J "" FIO,;S

J(h2 - h2 . )For an axial turbine at constant radius the work speed ~R = go + W2 ,IS (50)

be . 2parameter may wrltten as-

- If reheat is-neglected (since it is small) then the isentropic"' u efficiency of a turbine stage can be expressed as-1\ = (46)

(VOl - VO2)

for a stage having zero tangential velocity at exit- 11 = ~ + ~N V? + ~R w~ J -1 (51)

2u(VOI - VO2)u

>.. = -v;- (47) For a turbine stage with zero tan gen ti al velocity the effi-

01 ciency can be written in terms of the loss coefficients,

Th . t . b .tt ' f . f work-speed parameter, and axial velocity ratio, as-IS equa Ion can e rewrl en In terms o reactlon romequation 44 as- [ ] -1 2 1 VX ~N ~R>"

"' 1 ( 1 ) 11 = 1 + - >"(~N + ~R) ( - ) + - + - (52)1\ =""2 ~ (48) 2 u 2>.. 2

The value of work-speed parameter for an impulse (or zero SODERBERG'S LOSS COEFFICIENTS. Soderberg's

reaction) blade element is 0.5 and for a 50 per cent reaction correlations of loss coefficients, presented by Stenning (3)stage >.. is 1.0 when the leaving tangential velocity is zero. and Horlock (4) is in the form of a basic loss coefficient

TURBINE EFFICIENCY. The efficiency of a turbine which isafunctionoffluidturningangletogetherwithcor-can be predicted with reasonable accuracy from a know- rections for Reynolds number, aspect ratio, and tip clear-ledge of the geometry and velocity diagrmas at the mean of ance.the hub and tip radius of the machine. The various methods The basic loss coefficient, ~*, presented on Fig. 6, wasof prediction are based on correlations of losses for good evaluated for the following conditions:

designs. 1. Optimum solidity as given by Zweifel's rule, Fig. 9The loss coefficients are defined in terms of the differ- 2. Zero incidence

ence in enthalpy of the actual and an isentropic expansion 3. Reynolds number is 10sacross the blade row, Fig. 5. The nozzle loss coefficient is- 4. Aspect ratio is 3:1

5. Tip clearance is zero

J (h - h . ) Soderberg's correction factors for aspect ratio, Reynoldsl l IS .~N = go ' (49) Number, and tlp clearance can be expressed as-

t vt Aspect Ratio

67

( b ) AINLEY'S METHOD. Ainley and Mathieson (6) devel-~' = (1 + ~*) 0.975 + 0.075 - - 1 (53) oped an empirical method of predicting the performance of

h turbine stages at off-design as well as at the design point.

. ;. . The method is somewhat more complicated than the Soder-where b = blade axlal chord and h = blade helght berg method and uses pressure loss coefficients rather than

Reynolds Number enthalpy loss coefficients. The procedure takes account of

deviation angle leaving the blade row, secondary losses,

"-( 10S ) 1/4 , losses due to trailing edge thickness, and incidence effects.

~ - & ~ (54) This method will not be discussed here.

EFF ICIENCY PR EDICTIONS. The Soderberg methodwhere has been used to predict the isentropic efficiency at design

point for a turbine stage having zero tangential velocity atRe = DV I Iv, D == 2hs cos al/(h + s cos al) exit. The variation of isentropic efficiency with work-speed

f I d parameter and fluid angle leaving the nozzles is shown onor nozz es an F' 8 Th " f " h h ff ..' kIg.. IS Igure s ows t at e Iclency Increases as wor

R - DW 1 D = 2h l a I I(h I .a 1 ) speed parameter is increased and is maximum when thee - 1 V - S COS'-2 + s COS .-2 fl .d I I . h . ., UI ang e eavlng t e nozzles IS approxlmately 70 degrees.

for rotors. liI

o.

ro .09>.

~ Re = 105o.. !! Tip clearance = O. ~ 08 h/b = 30

: 0'2 ~2 = Oi "~ ~I -uo ~ O,. ~I

~of O~o 1'0 1-2

Fig. 6. Soderberg's Work -speed Parameler, >-. Loss Correlation.{Stenning (3) Cour- Fig. 8. Predicted Isentropic Efficienc~ uSi~g Soderber~'s Methodt M I ' tit (Courtesy Northern Research and Englneerlng -Corporatlon)esy ass. ns ute

o .. .0 lO lO '00 , ... of Technology)... O."..ti~. ,--¡

Smith (7) provided a correlation of many turbine-test re-TIP CLEARANCE. Soderberg's correction for tip clear- sults, all corrected to zero tip clearance. The correlation,

ance was to multiply the calculated efficiency by the ratio shown on Fig. 9 indicates that stage efficincy is a functionof the flow area minus the clearpnce area to the flow area. of work coefficient, (gol Llho IU2) and flow coefficient,This method of tip clearance correction is considered ade- (Vxlu),quate only for small impulse turbines and a more plausible OFF-DESIGN PERFORMANCE. It has already beencorrection factor from Cardes (5) is presented on Fig" 7. suggested that Ainley's method can be used to predict theThis correction takes account of reaction of the stage in off-design performance of turbine stages. Whitney andaddition to the tip clearance. Stewart (8) also developed a procedure for calculating the

performance of turbine stages at off-design conditions. At100 an angle of inciden~e, ¡, it was suggested that there was an

~ additional enthalpy loss given by-o" 098~ W2sin2¡~ Incidence loss = I (55)

r. 0"96 2gol2~ 0.91, Despite the apparent simplicity of this expression the pro->. cedure for calculating the off-design performance of a tur-~ bine stage is time-consuming, mainly beca use of the itera-:g 0,92 tion procedure to salve the continuity equation at outlet

W from the rotar blade row.0-90 LEAVING LOSSES. The kinetic energy of the fluid leav-

ing the turbine can be an appreciable portion of the avail-able enthalpy drop and it is desirable to minimize this

o 0"4 08 1"2 1.6 2.0 24 28 H .. f fClearance Ralio, k/h per c.nt energy loss. owever, the beneflclal ef ect on per ormanceof reducing the leaving axial velocity is offset by an increase

Fi9: 7. Tip Clearance Correction Factor (Cordes (5) by permission in the centrifu gal stresses in the rotar blades due to theSprl nger- Verlag)longer blades. The turbine performance often has to be

Soderberg's method can be used to predict the efficiency compromised in order to obtain satisfactory stress levels.of a turbine stage at design point. ~t is claimed that the pre- An efficient diffuser downstream of the last rotor is adicted efficiency is within two per cent of measured values. useful method of enhancing turbine performance, if space

! 68I¡

permits, The influence of diffuser recovery factor, e on the The radial equilibrium equations are derived from thetotal (or stagnation) to static efficiency of a turbine can be equations of motion* and basic thermodynamic relation-determined from the following equation: ships and can be applied to any turbomachine where the

-1 main direction of through flow is axial. This restriction to[ ¡,N ~ + ¡, W2 + (1 - e) V 2 + VO2] mainly axial flow appears because equilibrium in the radial1 t; I t;R2 x2 2 (56) d '" ' dd ' f ' l ' 11ts = + .. Irectlon IS consl !:!re at a series o axla statlons,

2ú(Vel - VO2) It is customary to make certain assumptions about the

flow in arder to simplify the analysis, The flow is generallyWhere the recovery factor e defines the proportion of the assumed to be axisymmetric, that is, circumferential varia-kinetic energy of the axial component of velocity recovered tions in fluid conditions are neglected, A further assump-in the diffuser, tion that is often used in axial turbines is to neglect the

SHROUDED ROTOR BLADES, It has been found that radial components of veTocity. Procedures using the lattertip leakage is an important source of loss in a turbine. Fig- assumption are termed simple radial-equilibrium methods,l.ire 7 shows that there is between 1,8 and 3,5 per cent loss In many turbines the radial components of velocity arein efficiency for one per cent of tip clearance, depending on small and can be neglected so that the more time-consum-reaction, One method of reducing the loss is to use ing, but more accurate, analysis which includes the effectsshrouded rotar blades in arder to reduce leakage flow by an of the radial components of velocity is not often necessary,appreciable factor without reducing the radial runningclearances, There is little published information on the de- 3.0sign of rotar shrouds, .~- 2,8

REYNOLDS NUMBER EFFECTS, Soderberg's correc- ~tion for Reynolds number given in equation 54 indicates ~ 2.2that the loss coefficient varíes asRe-%; Reynolds numbers ¡" 1,8were based on hydraulic mean diameter and velocities at ~ 1.'the throat, Ainley (9) found that from turbine cascade tests ~ Fig. 9, Turbinethe losses varied as Re-y., losses deduced from a four-stage ~ 1.0 Stage Efficiency

, ,'" ~ based on Smith'sturblne tended to conflrm thls result, Alnley also stated .6.3,' ,5 ,6 ,7 ,8 ,9 1,0 1,1 1,2 1.3 Correlation (7),that there was little decrease in loss when the Reynolds FI.w C.etticie.l. V,/U

number exceeded 1 x 105 for the four-stage turbine; thebasis fQr the Reynolds number was blade chord and outlet RADIAL EOUILIBRIUM (INCLUDING STREAMLINEvelocity, In a more recent paper by Holeski and Stewart CURVATURE), The equation of motion in the radial direc-(10) the influence of Reynolds number on the losses in sin- tion, equation 29, may be combined with the entropy equa-gle stage turbines was studied, Reynolds number was de- tion, equation 7, to eliminate the pressure gradient term tofined as: m!.urm-where m is the mass flow and rm is the give- -radius of the mean section of the blades, The main observa- ( as ah) a V a V ~tions from the analysis were that below approximately 2 x gol T - -- = Vr ~ + Vx -2- - ~ (57)105 there is a variation of loss with Reynolds number and ar ar ar ax rabove that Reynolds number there appeared to be verylittle variation of loss with Reynolds number, There was Usíng the definition of stagnation enthalpy, given in equa-considerable scatter in the variation of loss with Reynolds tion 3, the previous equation can be rewritten as follows tonumber although Re- % appears to be a more accurate rep- provide the radial equilibrium equation:

resentation than Re-y. '. Many of the turbines considered by ~. ) V 2Holeski and Stewart were small impulse turbines which are gol T~ - ~ = V 3-r - V !!!-= - Vo ~ - ~unrepresentative of gas turbine practice, ar ar x ax x ar ar r

The variation of loss wit Reynolds number in turbine (58)blades is complicated becau of the varying extents oflaminar boundary layer, In hi h re ct 'on t rb ' l . The first term on the right hand side of this equation isa I u Ines amlnar II h d ' ff ' I d . b "

1boundary layers might extend ver considerable regions of genera .v t e m?st I ICU t to eter,mlne e~ause It Invo ,ves

the blade Surfaces e en at h ' h cid b d h ' h the radial veloclty, V r' The evaluatlon of thls term requlresv Ig eyn s num ers an Ig h d " f h ' , f h l " hlevels of turbulence because of t e accelerating flow. Low t ,e pre Ictlon o t e,posltlons o t e stream Ines In t ~ tur-reaction blades would tend to hav mainly turbulent boun- bln~ and the calculatlon of the cur~atures (or second dlffer-dary layers, at least on the suctio surfaces of the blades entlals) and slopes of these streamllnes, .because of the adverse gradients ere, It is concluded SIM:LE RADIAL EOUILI,BRIUM,Theassump~lon~~attherefore, that it is impossible to gene lile about the varia- the ra~lal5c80mponent of veloclty can be neglected slmpllfles

, f I ' h R Id b ' h " l . equatlon to-tlon o oss Wlt eyno s num er sin t IS IS strong y In-fI,uen~ed ,bY ~he types of blades and the etails of the velo- ( ds dho) dV x d V e ~Clty dlstrlbutlons on the blade surfaces, gol T - - - = - V x - - Ve - - -

\ dr dr dr dr rII (59)

Radial Equilibrium /" For design purposes it is often assumed that the stagnation

The general term "radial equil~~ is used to define enthalpy and entropy are constant radially so that equationthe analytical method of calculating the radial variation in 59 reduces to-axial velocity at any axial station in the turbine, The value :,., ,

f' " " *The absence af bady farces In the equatlans af matlan Implles thatO axlal veloclty IS, of course, requlred as an Important the analysis strictly applies ta regians between blade raws and nat

component in the velocity triangles, within blade raws.

69

'.~._-

dV dVO Ve tion enthalpy downstream of the rotar and equation 59 is

V x -2- = - Vo - - - (60) used to determine the distribution of axial velocity thus-d, d".' . dV dh

This differential equation is easy to salve when simple varia- V x -2- = gol --.!!-. (69)tions in tangential velocity with radius are specified. d, d,

FREE VORTEX DESIGNS. The free vortex distribution

of tangential velocity, defined as- and it can be shown that-

;,VO =constant (61) [ ( ) COS2a ]V 2 V 2 x2 - xl '1 =u.V.l-- (70) is very easy to salve using equation 60. The solution indi- 2 II 011 '1 ¡

cates that the axial velocity, V x' is constant at all radii.

For a free vortex turbine stage where the inlet tangential The variation in axial velocity at exit from the turbine with

velocity is given by- radius can be determined from equation 70, It can be seen

from this equation that the variation in tangential velocity

v: = ~ (62) at ~xit from the tur.bine is presented i~ t~rms of t~e inlet01 , radlus 'l. However, In most cases there IS Ilttle loss In accu-

racy when the outlet radius '2 is used in this equation,

and the exit tangential velocity given by-

Turbine Blade Design

v: = !!.. The aerodynamic design of turbine blades is often close-

02 , (63) Iy linked with the mechanical design and also with the de-

sign for manufacture. The aerodynamic designer therefore,

the stagnation enthalpy drop is therefore- does not often have complete freedom to select the opti-

mum blade geometry. Despite this fact it is often possible

u(VO - VO) <v(a - b) for the blade designs to be clase to the optimum. A useful

~ho = I 2 = (64) design tool is an analytical method of calculating the veloci-

gol gol ty distributions on the blade surfaces. The effect of design

changes and compromises can then be assessed.

and the degree of reaction is- , Tests on a related series of blade profiles have been

- carried out fo( a small range of conditions but these data doVOl + V02 a + b not appear to be used in many turbine designs. Turbine

6{ = 1 - 2 = 1 - 22 (65) blades are normally designed by selecting the solidity byu <v, one or more of the empirical rules given below and then

E . 64 h h h h I d . drawing the blade shapes. When analytical procedures for

quatlon s ows t at t e ent a py rop IS constant at . . ,. , .

11 d .. f f d . 65 . d . determlnlng the veloclty dlstrlbutlons on the blade surfaces

a ra II or a ree vortex stage an equatlon In Icates. ...

h h d f .. . dl . h d . are avallable these would be used to modlfy the shapes, If

t at t e egree o reactlon Increases rapl y Wlt ra tus. t . th d .. necessary, o Improve e eslgn.CONS:ANT NOZZLE ANGLE, DESIGNS. ~any axlal- SELECTION OF BLADE SOLIDITY. There are two

flow turblnes have free vortex deslgns but occaslonally, for 11 k h d f I t . th I . d . t f t b .. ... we nown met o s o se ec Ing e so I I Y o ur Ine

manufacturlng purposes, It IS deslrable to use a constant bl d .

nozzle-angle designo The distribution of tangential velocity a l es Z. . f 1,

th d. I . . b . . wel e s me oat In et IS glven y- 2 A . l '

th d. In ey s me o

v: - V 66 These methods are both based on the analysis of experi-81 - x tan al () mental results and do not normally provide identical values

. . . . of optimum solidity. In the absence of additional informa-When thls dlstrlbutlon of tangentlal veloclty IS substltuted t . . t . b bl th f t d t I t th h ' h.. . . . .. Ion I IS pro a y e sa es proce ure o se ec e Ig er

In equatlon 60 the followlng dlfferentlal equatlon IS ob- f th t I f ' .d .t. o e wo va ues o so I I y.

talned: ZWEIFEL'S OPTIMUM SPACING. Zweifel (11) ob-

dV I d, served that the losses from turbine blades were approxi-~ = - sin2al - (67) mately a minimum at a particular value of tangential force

V xl' coefficient, 1/1. Tangential force coefficient is defined as-

This equation can be integrated to give- s

1/1 == 2cos2(32 (tan (31 -tan(32)- (71)sin2a1 b

V ( 'l. )~ = -!- (68) where b is t~~ width of the blade row and s is the

V xl ¡ '1 circumferential spaclng of the blades.

The optimum value of tangential loss coefficient is ap-where the suffix i refers to conditions at an initial radius '¡. proximately 0.8. Curves for determining the optimum spac-

The distribution of axial velocity downstream of the ing of turbine blades for a value 1/1 = 0.8 are given on Fig.

rotar depends on the desired variation in tangential velocity 10.

downstream of the rotar. For the cáse of zero tangential AINLEY'S OPTIMUM SOLIDITY. Ainley and Mathie-velocity after the rotar there is a radial variation in stagna- son (6) provided curves of loss coefficient against pitch-to-

70

2- . where

2- ar = cos-1 (;.)2-

1- 113; I = cos-1 (-;)J)- 1lO

~ O = throat width ~

.3 1~ s. = blade pitch corrected for exit blade thickness

¡¡~ 12 An alternate definition of deviation is used when the tur-CJI

~ bine blades are drawn as airfoils by defining the mean lines~ 10 and thickness distributions. Deviation for airfoil blades is

'"

measured from the mean line at or near the exit of theo- blade.

SUBSONIC DEVIATION. Experimental correlations ofdeviation data for turbine blades have been given by Ainleyo- and Mathieson (6) and Lee (12). In both correlations the

deviation angles are plotted against Mach number with exito angle as para meter and in the regions of interest the two

50 70 80 correlations agree. The correlation of deviation angles dueRelative Exi! Fluid Angle .-132 (deg) to Lee but replotted by Cordes (5) is presented on F ig. 12.

Fig. 10. Optimum Spacing. Zweifel's Method

chord ratio for nozzle blades ({31 = O) and impulse blades 7.

({31 = - 132) for a range of exit angles, and suggested that

the foss for other blades would vary as the square of the 6.fluid inlet angle. The pitch-to-chord ratios for minimumloss were calculated from these data by a curve fitting tech-nique for a wide range of inlet and outlet angles and are 5,presented on Fig. 11-,- ~

Q13 .."O~ 4-0

<..o1 .

ca~ 3-0

tiI --

>..

o

1 2.0u"Ui

..; Og~~ ,. 1-0uCJIc 08Ü..o. O

'"

0.4 0-5 0-7 o-e. oExit Mach Number

o Fig. 12- Subsonic Oeviation (Cardes (5), by permission Springer-Verlqg)

O.SUPERSONIC DEVIATION. When the flow down-

stream of the throat becomes superconic there are patterns50 60 70 of shock waves and expansion waves which turn the average

Relaliv~ Exi! Fluid Angl. -13 (deg) . . . d .. D . h (13) d. 2 flow and glve rlse to supersonlc evlatlon. elc es-

Fig. 11. OPtimu~ Spacing, Ainley's Correlation (Caurtesy Northern cribed an experimental investigation of supersonic condi-Research and Englneerlng Corporatlon) .

d t f th bl d d f d th t th pertlons owns ream o e a es an aun a e su -DEVIATION D . t . . th t d t d .b h sonic deviation angle could be predicted accurately pro-. evla Ion IS e erm use o escrl e te. . . .

fl . d . h t k I d f h h f vlded the tralllng edge thlcknesses were small- It can also be

UI turnlng tata es pace ownstream o t e t roat o . . .

th b I d D . . 11 d d h fl .d seen from hls results that the devlatlon angle at low super-e a e. evlatlon genera y ten s to re uce t e UI. . -t . . th bl d d - d f . d sonlc Mach numbers can be predlcted very accurately uslngurnlng In e a e an IS e Ine as- -. - . .

j the contlnulty equatlon and the assumptlon that the flow IS8 == ar - al for a nozzle blade row isentropic from the sonic throat to the downstream condi-

and (72) tions. Supersonic deviation angles predicted using the isen-8 == 132 - (3; for a rotor blade row tropic assumption are presented on F ig. 13.

71

)-0 dV V- = - (73)dn rc

where n is measured in the normal direction and r c is theradius of curvature considered positive when the centre ofcurvature is in the direction of increasing n.2-0 This equation provides a method for calculating the flow

condition along normal~ to the streamlines. An iterative! solution is normally used for determining the flow condi-~ tion in a channel. The continuity equation is first used to.., provide the positions of the streamlines for the calculation

; of the radius of curvature of the streamlines in equation 73.

~ 1.0 This method can be used in compressible as well as incom-

-~ pressible flow and in addition it is not restricted to two-~ dimensional channels since allowance can be made in con-

tinuity calculation for any change in spacing of the stream-lines in the axial planeo

An elegant solution of the cascade problem was devel-o oped by Martensen (15). His process is essentially the solu-1-0 105 1.1 1-15 1.20 1-25 tion of an integral equation describing the distribution of

Eail Nach Numb.r velocity on the blade surfaces. The method is suitable for

Fig. 13. Supersonic Deviation Angles for Turbines Blades (1 = 1.33, solution using a digital computer and can be modified toR = 53.3 ft Ibf per Ibm deg R) salve compressible flow problems, Payne (16).

The channel flow methods are relatively easy to salve. THEORE~ICAL BLADE DE?IGN MET~ODS. The d~- using a digital computer and much can be gained in terms

sl~n of turb.lne blade sha~es .uslng theoretlcal methods IS of performance or perhaps confidence in performance bywld.ely used In the gas. turblne Industry. Several methods a!e proper use of these methods. The more elaborate methodavallable .for the solutlon of the channel flow problem whlle of Martensen is somewhat more difficult to salve using aothers can be used for solving the cascade flow probler:n. digital computer but the results would be more accurate,

In channel flow methods the problem of the flow In the particularly in the regions of the leading and trailing edges.cascade is simplified by considering only the flow betweenone pair of blades. These metlteds are not accurate in the CASCADE TEST RESULTS. An investigation of a re-regions of the leading and trailing edges of the blades. The lated series of turbine-blade profiles in cascade was con-principie of the channel methods is illustrated on Fig. 14. ducted by Dunavant and Erwin (17). Low speed tests were

Stanitz (14) described an analytical procedure for calcu- carried out on profiles having a range of camber angles fromlating the shape of the turbine channel having prescribed 65 degrees to 120 degrees. Solidities of 1.5 and 1.8 werevelocity distributions on the blade surfaces. The method used and each section was tested over a practical range ofcan be used for incompressible and compressible flow in inlet angles. Some tests were also carried out at high speedtwo dimensions. However, a relaxation method must be conditions to determine the values of critical Mach number.used for compressible flow and this is time consuming to Design incidence was determined from the measurementsalve on a digital computer. A good approximation using of pressure at a tapping located at the leading edge of eachlinearized compressible flow is also provided by Stanitz blade. When the pressure at the tapping registered the stlg-which is very easy to salve using a digital computer. nation pressure then this was defined as the design inci-

dence (Dunavant and Erwin used the term induced angle).These tests showed that the design incidence was negativeand a function of inlet angle and solidity; varying from -17degrees at zero inlet angle, solidity 1.5 to -2 degrees at 60degrees inlet angle, solidity 1.8.

,--( 5 ., uctlon, ,- 5 fI 010 I \ ur ace """ ,~ \

\ I\\ ,\, \

", \", \ b" ,

" "Fig. 14. Channel " "Flow between Tur- "bíne Blades " "

"Streamline curvative methods can be used to determine

the velocity distribution of channels of prescribed geome- S'try. The basic equation for this method is derived from the Fig. 15. Blade De-condition for irrotational flow- sign Procedure S

72

BLADE DRAWING PROCEDURES. An examination of '9

turbine profiles used in curre,lt gas turbines shows that ,8there are three main features of blades and channel shapes: '7

1. The blade turning is highest near the leading edge2. The suction surface of the blade is nearly straight L~Q.° .6

from the throat to the ti"ailing edge. ~ ,5

3. The geometric throat of the blade is at exit from the ~- -4blade channel or passage. ...

These features are therefore used in the preliminary design : ,3

of the blade channels and profiles together with any stress- ; 1'0ing requirements for maximum thickness and trailing edge .thickness. The drawing procedure is illustrated on Fig. 15.

THE RADIAL INFLOW TURBINE .,Turbine Performance

TURBINE CHARACTERISTICS. If dimensional analysis .6

is used it can be demonstrated that dynamical similarity is >-

maintained in a turbine with varying inlet conditions pro- ~vided (m~ IPo 1), (u31 ~l) and Reynolds number re- :~ '4

main constant. Experiments show that the performance is ;less sensitive to changes in Reynolds number than to .changes in the other two parameters. The effect of changesin Reynolds number is therefore considered as a separa tefactor and is not generally presented in the turbine charac- ,2.4.'.8 1.0 1.2teristics. Although characteristics for radial inflow turbines v.. ' A . / ,.- .. ocII)' allo, u, -i&GeJan can be drawn in the same form as axlal-flow turblne charac- o

teristics Fig. 3 there is a tradition for presenting them in a Fig. 16. An .Example of a Radial-lnflow Turbine C~ara.cteristlc, , . (based on Hlett and Johnston (20), Courtesy I nstltutlon ofdifferent way. This appears to have developed from Impulse Mechanical Engineers)and steam practice where the efficiency is plotted against

blade-jet speed ratio, (u3/V2goJ¡),ho,i-r). In this expression meet the blades at the leading edges at atangential velocity¡),ho,;-r is the isentropic enthalpy drop from inlet stagnation given by slip conditions. At this operating point there areto exit stagnation (or sometimes static) conditions. An ex- no velocity peaks at the leading edge and this suggests thatample of a turbine characteristic drawn in this way is shown inlet losses to the rotar would be a minimum. However, inon Fig. 16. The isentropie-efficiency characteristic for this arder to obtain maximum efficiency, the tangential velocityturbine is nearly unique for stagnation pressure ratios be- leaving the rotar should be zero at this condition.tween 1.3 and 2.45. The stagnation-to-static efficiencycharacteristic varies slightly with stagnation pressure ratio Station Numberover the same range. The mass flow characteristic is also

. . Collectorshown on thls figure.

VELOCITY TRIANGLES. The components of the radialinflow turbine together with the station numbers are illus- otrated on Fig. 17 and the velocity triangles at inlet and exitof the rotar are shown on Fig. 18. The velocity triangles at a

exit of the rotar are often drawn at a station corresponding P

to the trailing edge of the blades so that the blockage of the 01trailing edge thickness is accounted for in the continuity -

equation.WORK EQUATION. The stagnation enthalpy drop Fi9.17.ComponentsofRadlal-lnflowTurbine

across a radial inflow turbine along any streamline can be~ ~~~~~~~~derived from equation 28 as-

~3U3 VO3 - U4 VO4 IJ (-)ft;:~ hO3 - hO4 = (74) 3 V3

gor W3 Vm3

In many applications it is desirable to minimize the kinetic I ~ ve3 --jenergy leaving the turbine so thát the tangential velocity I "3 --1leaving the machine is arranged to be zero. Equation 74h d a Inl.1 Triangl.t en re uces to- -

11'144(-) U3 "03 hO3 - hO4 = (75) ~4

golW4 Ym4 V4

OPTIMUM INCIDENCE. An examination of the stream-lines at the inlet to the rotar of a radial inflow turbine ~ " .:J:!t-shows that there is a striking resemblance to ffow condi- 4

tions in a centrifugal compressor; this is indicated on Fig. ~ Exil Triangl.

19. The stagnation streamlines for the radial inflow turbine Fig. 18. Velocity Trlangles for a Radial Infiow Turbine

73

.'.0

.'ion

0.9

0.8 -~ Hielt and 10hnslonc. Based on Average:~ Slalic Density;; 0.7 . I I

Bas'ed on Inlel SI.gn.lio~

Densily0.&

0.5O 0.05 0.25 0.30

Sp&cific Sp&&d, Nsa COMPRESSOR b. TURBINE- - - Fig. 20. Correlation of Peak Efficlency with Specific Speed

Fig. 19. Comparislon of the Streamlines for the 1 mpeller of a Cen-trifugal Compressor and for the Rotor of a Radial-lnflow Turbine W. l. (21) f . f . I fl t b.IS Icenus trans ormatlon rom an axla - ow ur Ine

The stagnation enthalpy drop at the optimum incidence ca~ca~e. With these nozzles the solidity, defined as chord/. h . 11 ' I 't' exlt pltch,wasselected as 1.33.

Wlt zero tangentla eavlng ve OCI YIS- SUBSONIC DEVIATION. Hiett and Johnston observed

( VfJ3 ) that there was very good agreement between the measuredu~ - fluid angle downstream of the nozzles and the fluid angle

Dho = U3 opt (76) predicted using the Wislicenus transformation. They also

gol found that the approximate rule expressed as-o* -1where a2 = cos -¡ (80)

( VfJ3 ) = Vm3 (3* + r . . - - tan 3 ~ (77) gave reasonably good agreement wlth the experimental re-

U3 U3 sults. In this equation s is the circumferential pitch.Opt SUPERSONIC DEVIATION. In the design of radial-in-

(33 is the blade angte atTotor inlet flow turbines for very high pressure ratios it is often neces-t is the slip factor for centrifugal compressors given by sary to have supersonic conditions downstream of the noz-

equations 81,82, or 83 of Chapter 3, of this Handbook. zle exit. Although convergent-divergent nozzles could beFor a rotar with radial blades at inlet (33 is zero and the used for this purpose it has been found that a more accept-

stagnation enthalpy drop is- abre design solution in many cases is to use convergent noz-

tu2 zles and allow a supersonic expansion downstream of the~ho = -2 (78) nozzles. If sufficient radius is available a shockless expan-

gol sion can be provided by a vaneless space; but there may belarge friction losses. An alternative design procedure is to

and the slip factor, t, is generally clase to 0.85. arrange that the expansion from sonic conditions at theSPECIFIC SPEED. Specific speed is a non dimensional throat to supersonic conditions downstream of the nozzle

parameter defined as- occurs through a series of shocks and expansions at the

trailing edge of the nozzles. This method would be utilizedN ..¡Q when a restriction is placed on the maximum radial dimen-

Ns == 601g 1 ~ h )% (79) sion. The shock losses associated with the supersonic e.xpan-o o sion are very small when the leaving Mach number IS low

. . «1.25). Supersonic deviation can be calculated assumingwhere Q IS the 1~let volu":,e flow.. . . . the flow is isentropic from the sonic throat. The method is

For conventlonal deslgns of turblnes, speclflc speed IS f 1I .f h d . f as o ows.closely related to the ~eometry o t e rotar a~ IS r.e~uent- At a supersonic Mach number equal to M2 at outletIy used as a correlatlng para.meter f~r tu~blne e.fflclency. from the nozzles the fluid angle is a2 and the blade angleThe performance. of h~draullc machlnes In particular ha.s iven b cos-1 (o/s) is a;. a2 and a; are related by the

been correlated wlth thls parameter, and the value of specl- 9 .y

f . d . . d . f W d (18) expresslon-IC spee IS a very Important eslgn eature. 00 pre-sented a correlation of peak efficiencies for radia! inflow (m~ )turbines as a function of specific speed. Wood's correlation o

together with add itional data are presented on F ig. 20. cos a; P oA M = M 2- = - (81)

Component Design cos ~2 (m VT o )The terminology used to define the components of a -~A M = 1.0radial inflow turbine is given on Fig. 17.

NOZZLE BLADES. The design of nozzles for radial in- where (m V-r:;;-/PoA) can be calculated for a perfect gasflow turbines was investigated by M i?umachi (19) but the using equation 24.results were inconclusive. Hiett and Johnston (20) designed VANEL.ESS SPACE. Between the nozzles and the inletthe nozzles for angles of 60, 70, and 80 degrees using the to the rotar there is normally a gap termed the vaneless

74

space. This gap may be utilized to provide an increase in centrifugal compr8Ssors. These methods are generally quasi-

Mach number from sonic conditions at the throat. Normal- three dimensional methods which use the assumption that

Iy, however, the gap is small and chosen for mechanical or the flow conditions can be determined in two stages as fol-

manufacturing considerations. Flow in the vaneless space lows:

can be analyzed by one-dimensional methods allowing for 1. Axisymmetric flow obtained by rotating the mea n

wall friction in a way similartd the analysis of vaneless dif- line of the blades around the rotational axis of the

fusers for centrifugal compressors. In many designs where machine.

the vaneless space is small the friction can be ignored and 2. Blade to blade flow obtained by using the blade load-

the change in flow conditions can be calculated assuming ing effects together with the mea n conditions given

isentropic flow together with the assumption that moment by the axisymmetric solution

of momentum, ,Ve, remains constant across the vaneless The equation defininp the gradient in relative velocity

space. along normals to streamlines in the hub-to-shroud plane was

ROTOR. The rotar of a radial-inflow turbine is a compli- given by Smith and Hamrick (24) for rotors with radial ele-

cated three-dimensional shape and it follows that the flow ment blades as-

conditions relative to the rotar are difficult to analyze. For ~ = W - b (83)preliminary design purposes it is often assumed that the dn a

meridional velocities at inlet and exit of the rotar do not

change from hub to shroud. where n is measured along each normal and a and b are fac-

NUMBER OF ROTOR BLADES. Using the assumption tors which depend on the geometries of the streamlines and

that the flow at inlet to the rotar does not vary from hub the mean blade surfaces.

to shroud and the condition for absolute irrotational flow The values of the factors a and b are given by-

Jamieson (22) obtained an expression for the minimum

number of blades to prevent reversed flow at the blades. 2 a . 2 aTh .. b f bl d d . cos.. sin..

e mlnlmum num er o a es can be etermlned from- a = - - -

'c , cos e- 21T cos 2 (33 (84)

Z - (82) sin (3 ( dWe )Vm3/U3 b = - sin e cose - + 2(,,)

., ... . cose dxJamleson s analysls IS approxlmate but appears to predlct

blade numbers which are Glose to the optimum number for .. .radial inflow turbines in some cases. Hiett and Johnston As In oth~r streamllne curvatu!e methods the r~dlus of

(20 ) measured a sli g ht im p rovement ( onet) ' ff '- curvature 'c IS normally the most Important factor In deter-

per Gen In el.. h . - . l . l .I h Iciency by doubling the number of blades to conform to mlnlng t e v~rlatlon In re a~lve ve oclty a ong t e norma ~.

Jamieson's mínimum value at a high nozzle angle setting. . Fo~ machlne~ ?f prescrlbed geQ~etry t~e flo~ con~l-Knoenschild( 23 ) discussed modifications to a low '- tlons In the meridional plane are obtalned uslng an Iteratlve

specl d . h . h h . . f h . dtic speed turbine to improve the distribution of rel t" proce ure In w IC t e posltlons o t e streamllnes are a -

velocities on the blades of the rotors. This procedu~elvi~ justed until the continuity equation and equation 83 are

illustrated on Fig. 21. simultaneously satisfied. Smith and Hamrick (24) devel-

oped a simple approximate method which was not iterative

Improved but simply a step-by-step integration from known initial

c Original condítions. With this method the shape of an initial stream-

~ \ line and the variation in relative velocity along the stream-É - - - - - - - line are prescribed. The procedure is then to work away ,

5~ IJproved Shape from this initial streamline by solving conditions across an~ incremental streamtube using the basic flow equations. In-c this way a series of streamlines are constructed from the

-- - - - 1'0. hub to the shroud. If the shape or velocity distribution on

Diamolor I Tip Oiamoler the final streamline is unacceptable then a new distribution

of velocity along the initial streamline is assumed.

The blade to blade conditions can be determined by the

~ !la ..~ simple approximate method of Stanitz and Prian (25). With '- u ~ 2 ~ ~ this method the concept of zero circulation is used together

:~mloan : : with simple assumptions relating the blade surface velocity

: ~: to the average velocity. At optimum incidence the velocity. - ~

:: ~:: distributions near the leading edge of the rotar blades canbe determined using the assumption that the flow at opti-

Mean Slreamline mum incidence is analogous to slip flow in the impellers oflenglh centrifugal compressors. With this assumption the flow in

a Original Shape b Improved Shape c Increa.e 0151ade Number the region of the leading edge is assumed to follow the

Fig. 21. Procedure for Improving the Performance of a Radial- direction given by-Inflow Turbine (Based on Knoernschild (23), Courtesy of Mechan-¡cal Engineers) 2( ' ) ( ' )sin(3=E+F - + G - (85)

THEORETICAL DESIGN METHODS. The flow condi-'3 '3tions in the rotar of a radial-inflow turbine can be analyzedusing methods developed for the design of the impellers of Where the constants E, F, and G are selected to satisfy the

75

boundary conditions- ROTOR SCALLOPING. Rotors of radial-inflow turbines

are often scalloped to lighten the wheel, also to prevent13 = 133 at, ='3 ;.; . cracking between blades. Penalties in efficiency found by

13 = blade angle at, = 'x H iett and Johnston for such modifications depended on thed(sin 13) = gradient of sin of blade angle at radius, extent of the scallops. Maximum loss in efficiency was 4

dr x per cent with rather deep scallops. This loss in efficiency

The flow is assumed to follow the blades at a radius given was reduced to 2 per cent by removing material at the backby- of the disc to reduce the sudden change in cross section. It

was established by thesé tests that there were penalties inIn ~ = -O. 71 ~ sin E (86) efficiency resulting from removal of the disc between

'3 z blades but the penalties were small if care was taken to min-imize discontinuity in the flow path between the blades.

Losses and Efficiency OFF-DESIGN PERFORMANCE. The off-design perfor-The isentropic efficiency of a radial inflow turbine can mance of a radial inf~ow tu:bine can be calcula~ed from a

be written in terms of the actual stagnation enthalpy drop, k.nowledge of. :he deslgn polnt losse~ t<:>gether wlth correc-~ h across the turbine and the "enthalpy" losses as- tlons for addltlonal effects due to Incldence on the rotar

o, blades and tangential velocity in the exhaust duct.

77 = ~ = ~ho (87) Futral and Wasserbauer (27) described a method for de-

~ hO.is ~ ho + ~ losses termining the off-design performance of radial inflow tur-bines based on the method of Whitney and Stewart (8) for

The overall "enthalpy" loss in the radia-inflow turbine axial-flow turbines. With this method the loss coefficientscan be separated into losses for the various components. were determined in terms of the mean of the inlet and out-These component losses can further be subdivided into jet kinetic energies.losses due to various causes, such as skin friction, tip clear- Stator Loss up to the Stator Exit-

ance and corner losses. Useful information on the losses inradial inflow turbines has been given by Jamieson (22) and Vl> + v¡Balje (26). Ll = Kl 2gol (90)

The plot of efficiency against specific speed presentedon Fig. 20 provided another very simple method of estimat- Loss from Turbine Inlet to Rotor Inlet-

ing turbine efficiency.Hiett and Johnstonj20) provide performance data for L =--K ~~ (91)

such factors as nozzle angle, clearances, Reynolds number, s 2g 1and scalloped rotors. This information is summarized o

below. Rotor Loss-NOZZLE ANGLE. Tests were carried out with one tur-

bine rotar at nominal nozzle angles of 60, 70, and 80 de- W2 + W~grees. The efficiency was a maximum at a nozzle angle of LR = mK 3 (92)

70 degrees and fell slightly at 80 degrees. However, the '}gol

rotar was designed for a nozzle angle of 70 degrees and hasbest matching at this condition. The results are somewhat The stator loss up to the stator exit, L 1, is required ininconclusive but it appears that both 70 and 80 degree noz- addition to the full stator loss up to the rotar inlet, Ls, be-zle angles produced higher efficiency than the 60 degree cause the moment of momentum is required to determinenozzle angle. the rotar work. It can be seen that the ratio of rotar loss

CLEARANCES. Basic performance tests on the turbines coefficient to stator loss coeffi~nt, m, is used, rather thanwere carried out with a clearance on the rotar between the assigning separate unrelated loss coefficients. Conditionsblades and the casing of approximately 0.01 in. (rotor were evaluated at the arithmetic mea n of the hub and thediameter 5 in.). Tests with varying clearance were carried shroud diameter at rotar exit.out on one turbine and it was found that an increase in For calculating the loss due to incidence it was assumed.clearance of one per cent of the exit blade height reduced as in the case of axial flow turbines, that the loss was equi-the efficiency by approximately one per cent. Losses in valent to the kinetic energy of the velocity component nor-efficiency due to nozzle clearances were found to be, to a mal to the blade at rotar inlet. For a rotar with radialfirst order, one per cent for each per cent of nozzle height. blades at inlet the incidence loss was expressed as-

REYNOLDS NUMBER EFFECT. Tests were carried outon one radial inflow turbine over a Reynolds number range W2 . 2 13

of 0.5 x 105 to 2.2 X 105. Reynolds number was defined Lin = 3 sin 3 (93)

as- 2gol

Re == ~ (88) It was assumed in this analysis that minimum loss occurredv with zero incidence. A more satisfactory method might be

. . to use the optimum conditions; equation 93 would then bewhere b IS the top wldth of the rotor. rewritten as-

Hiett and Johnston found that the change in efficiencycould be expressed in the form W~ sin2 (133 -133 opt)

Lin = (94)

1-77cx:Re-o.16 (89) 2gol

76

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Futral and Wasserbauer (27) analyzed the performance absolute inlet angle, al , 64.7 degof a commercial turbine having a 4.5 in. diameter rotar and inlet meridional velocity, V mi, 644 ft/seccencluded that the value of K I for the turbine nozzle was exit meridional velocity, V m2, 550 ft/sec0.078,Khad a value of 0.1995 and m was 2.215. A feature inlet blade speed, U¡, 1664 ft/secof the turbine design analysed was the use of circular noz- exit blade speed, U2, 1198 ft/seczles rather than vanes in the"stator. The stator loss, Ls, was ex it absolute angle, a2 , O degtherefore somewhat higher than one would expect for specific heat, cp 0.272 Btu/lbm Rvaned nozzles because of the sudden expansion after thecircular nozzles. Comparison between the predicted and Answers:measured off-design performance was good over wide al 64.7 deg; 131 -25.0 deg; 132 - 65.6;ranges of speed and pressure ratio. ",Vel 1364 ft/sec; ve2 O ft/sec; Wel - 300 ft/sec;

Acknowledgements We2 -1198ft/sec; VI 1510ft/sec; V2 550ft/sec;The assistance of the staff of the Drawing Office at the ~ho 91 Btu/lbm; ~ T o 334 deg R.

Mechanical Engineering Department of the Imperial Collegeof Science and Technology, London, in preparing the draw- 2. Derive an expression for the specific speed of a radial-ings for this chapter is gratefully acknowledged. inflow turbine in terms of the ratio of the inlet dia-

meter, di, to the outlet shroud diameter, d2, for thefollowing conditions:

EXAMPLESrelative angle at inlet, 131 , O deg

Axial-Flow Turbine Examples relative angle at shroud exit, 132, 60 deg

1. Construct velocity triangles (not to scale) for the hub hub diameter/rotor tip diameter 0.25or r.o°t regi~~ of an axial-flow turbine having the fol- absolute tangential veolcity atlowlng condltlons: exit, Ve2, O ft/sec

absolute angle, al , 70 deg A .nswer.axial velocity, V xl = V X 2, 500 ft/sec ~d2~3 1 ~d~blade speed, UI = U2, 800 ft/sec Ns = 0.214 - - - -

rotar deflection,.81 -132, 107 deg d J 16 dIspecific heat, cp 0.272 Btu/lbm R

Determine the drop in stagnation enthalpy and stag-nation temperature for this turbin~ REFERENCES -Answers: . ..

TT Axlal Flow Turblnesa2 O deg; 131 49.0 deg; 132 -58 deg; "e 1 1375 ft/sec;TT Of / . W 760 f / . W 944 f / . 1. Axial Flow Compressors, J.H. Hcirlock, Butterworths, London,"e2 tsec, 1 tsec, 2 tsec, 1958,p.7.~ho 44 Btu/lbm; ~ T o 162 deg R. 2. A Study of Axial-Flow Turbine Efficiency Characteristics in

. Terms of Velocity Diagram Parameters, W. L. Stewart,2. Calculate the absolute and relatlve Mach numbers for A . S . t f M h . I E . ee P pe No 61 WAmerlcan ocle y o ec anlca ngln rs, a r . - -

the turbine root section described in Example 1 when 37, 1961.

the inlet stagnation temperature T 01 = 1500 deg R, 'Y 3. Design of Turbines for High-Energy-Fuel Low-Power-Output

= 1.33, and R= 53.0 ft Ibf/lbm R. Applications, A. H. Stenning, Massachusetts Institute of Tech-

Answers: nology, Cambridge, MIT Dynamic Analysis and Control Labora-tory Report No. 79, 1953.

MI = 0.835; MI R = 0.435; M2 = 0.468; M2R = 0.885 4. Losses and Efficiencies in Axial-Flow Turbines, J. H. Horlock,

3. Derive the expression for the specific speed of an International Journal of Mechanical Science, Vol. 2, 1960, pp.axial-flow turbine in terms of the hub-tip ratio of the 48-75.

h. A . I ( t . ) d. . 5. Stromungstechnik der Gasbeaufschlagten Axialturbine, G.mac Ine. ssume Impu se zero reac Ion con Itlons

C d S V I B I. or es, prlnger- er ag, er In.at the hubo 70 deg nozzle leavlng angle at the hub, 6 A M th d f P f E t . t . f Ax . I FI T rb 'nes. e o o er ormance s Ima Ion or la - ow u I ,

uniform axial velocity, uniform work at all radii and D. G. Ainley and G. C. R. Mathieson, Aeronautical Research

zero tangential velocity at exit. Council, London, ARC R and M No. 2974,1957.

Answer: 7. A Simple Correlation of Turbine Efficiency, S. F. Smith, Jour-nal of the Royal Aeronautical Society, Vol 69, July, 1965 pp467-470.~~ 8. Analytical Inyestigation of Performance of Two-Stage Turbine 2

, t ayer a Range of Ratios of Specific Heats From 1.2 to 1 2/3, W.Ns = 0.143 V:...: \ - 1 J. Whitney and W. L. Stewart, National Aeronautics and Space\ '1 Administration, Washington, NASA TN O 1288, 1962.

9. Performance of Axial-Flow Turbines, D. G. Ainley, ProceedingsRadiallnflow Turbine Examples of the Institution of Mechanical Engineers, London, Vol. 159,

1 C h l . . 1 ( t 1 ) . I 1948, pp. 230-244.. onstruct t e ve oclty trlang es no to sca e at In et 10 S d f NASA d NACA S. I S A . I FI T b. . tu y o an Ing e. tage xla ow ur Ine

and exit to the rotar at the shroud region of a radial Performance as Related to Reynolds Number and Geometry, D.inflow turbine and determine the drop in stagnation E. Holeski and W. L. Stewart, Journal of Engineering for Power,enthalpy and stagnation temperature for the follow- Trans. ASME Ser. A, Vol. 86,1964, pp. 296-298.ing conditions: 11. The Spacing of Turbo-Machine Blading Especially with Large

77

~~,-~

Angular Deflection, O. Zweifel, The Brown Boveri Review, 20. Experiments Concerning the Aerodynamic Performance in In-December, 1945, pp. 436-444. ward Flow Radial Turbines, G. F. Hiett and l. H. Johnston,

12. Theory and Design of St.vam and Gas Turbines, J. F. Lee, Paper 13, Thermodynamics and Fluid Mechanics Convention,McGraw-Hill Book Company, Inc., New York, 1954. Cambridge, England, April 1964, Institution of Mechanical

13. Flow of Gas Through Turbine Lattices, M. E. Deich, National Engineers, 1964.Advisory Committee for Aeronautics, Washington, NACA TM 21. Fluid Mechanics of Turbomachinery, G. F. Wislicenus, McGraw-1393,1956. Hill Book Company, Inc., New York, 1947, p. 211.

14. Design of Two-Dimensional Channels with Prescribed Velocity 22. The Radial Turbine, A. W. H. Jamieson, Gas Turbine PrincipiesDistributions Along the Channel Walls, J. D. Stanitz, National and Practice, editor {>ir H. Roxbee-Cox, George Newnes, Lon-Advisory Committee for Aeronautics, Washington, NACA Rep. don, 1955.1115, 1953. 23. The Radial Turbine, for Low Specific Speeds and Low Velocity

15. Die Berechnung der Druckverteilung an Dicken Gitterprofilen Factors, E. M. Knoernschild, Journal of Engineering for Powermit Hilfe van Fredholmschen Integralgleichungen Zweiter Art, Trans ASME, Ser. A, Vol. 83, 1961 pp 1-8.E. Martensen, Max-Planck I nstitut fur Stromungsforschung, 24. A Rapid Approximate Method for the Design of The Hub-Gottingen, Mitteilungen Nr. 23, 1959. Shroud Profiles of Centrifugal Impellers of Given Blade Shape,

16. Isolated and Cascade Aerofoils, D. Payne, Mathematics M. Sc. K. J. Smith and J. T. Hamrick, National Advisory CommitteeThesis, University of London, 1964. for Aeronautics, Washington, NACA TN 3399, 1955.

17. Investigation of a Related Series of Turbine-Blade Profiles in 25. A Rapid Approximate Method for Determining Velocity Distri-Cascade, J. C. Dunavant and J. R. Erwin, National Advisory bution of Impeller Blades of Centrifugal Compressors, J. D.Committee for Aeronautics, Washington, NACA TN 3802, Stanitz and V. D. Prian, National Advisory Committee for Aero-1956. nautics, Washington, NACA TN 2421, 1951.

Radiallnflow Turbines 26. A Contribution to the Problem of Designing Radial Turboma-18. Current Technology of Radial-lnflow Turbines for Compressible chines, O. E. Balje, Transactions of the American Society of

Fluids, H. J. Wood, Journal of Engineering for Power, Trans. Mechanical Engineers, Vol. 74, 1952, p. 451.ASME, Ser. A, Vol. 85,1963, pp. 72-83. 27. Off-Design Performance Prediction with Experimental Verifica-

19. A Study of Radial Gas Turbines, N. Mizumachi, University of tion for a Radial-lnflow Turbine, S. M. Futral and C. A. Wasser-Michigan, Ann Arbor, Industry Program of College of Engineer- bauer, National Aeronautics and Space Administration, Wash-ing Report No.1P-476, 1960. ington, NASA TN 02621,1965.

- A.DOUGLASCARMICHAELMassachusetts Institute of Technology

Cambridge, Mass.

Dr. Carmichael graduated from London University in 1949 andobtained his Doctor's degree at Cambridge University in 1958.

He began his industrial career at Bristol Aero Engines Ltd. (IaterBristol Siddeley) and spent severa! years in the compressor group onthe desing of axial flow compressors. On his return from CambridgeUniversity in 1958 he beca me head of the aerodynamic researchsection responsible for the advancement of the design of both com-pressors and turbines.

In 1961 he joined Northern Research and Engineering Corpora-tion of Cambridge, Mass. as senior project engineer and was respon-sible for many programmes in the field of turbomachinery. He wasone of the directors of the extensively sponsored research pro-grammes for the design of centrifugal compressors and radial inflowturbines.

In 1964 Dr. Carmichael became a Research Fellow in Turbo-machinery at the Imperial College of Science and Technology inLondon, and in 1968 he joined the English Electric Company astechnical Consultant.

In 1970 he was appointed Professor of Power Engineering in theDepartment of Ocean Engineering at the Massachusetts I nstitute of

Technology.

78

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